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Tomislav Ivezi´c Rud-.15em[1.25ex].2em.04ex-.05emer Boˇskovi´c Institute, P.O.B. 180, 10002 Zagreb, Croatia [email protected]

In this paper it is exactly proved that the standard transformations of the three-dimensional (3D) vectors of the electric and magnetic fields E and B are not relativistically correct transformations. Thence the 3D vectors E and B are not well-defined quantities in the 4D spacetime and, contrary to the general belief, the usual Maxwell equations with the 3D E and B are not in agreement with the special relativity. The 4-vectors Ea and Ba, as well-defined 4D quantities, are introduced instead of ill-defined 3D E and B. The proof is given in the tensor and the Clifford algebra formalisms.

PACS numbers: 03.30.+p

It is generally accepted by physics community that there is an agreement between the classical electromagnetism and the special relativity (SR). The standard transformations of the three-dimensional (3D) vectors of the electric and magnetic fields, E and B respectively, are considered to be the Lorentz transformations (LT) of these vectors, see, e.g., [1], [2] Sec. 11.10, or [3] par.24. The usual Maxwell equations (ME) with the three-vectors (3-vectors) E and B are assumed to be physically equivalent to the field equations (FE) expressed in terms of the electromagnetic field tensor F ab. In this paper it will be exactly proved that the above mentioned standard transformations of E and B are not relativistically correct transformations in the 4D spacetime and consequently that the usual ME with E and B and the FE with F ab are not physically equivalent. The whole consideration will be mainly presented in the tensor formalism (TF), since it is better known, and only briefly in the Clifford algebra formalism (CAF). It will be shown that in the 4D spacetime the well-defined 4D quantities, the 4-vectors of the electric and magnetic fields Ea and Ba in the TF (as in [4,5]), or, e.g., the 1-vectors E and B in the CAF (as in [6]), have to be introduced instead of ill-defined 3-vectors E and B. Let us start with some general definitions. The electromagnetic field tensor F ab is defined without reference frames, i.e., it is an abstract tensor, a geometric quantity; Latin indices a,b,c, are to be read according to the abstract index notation, as in [7] and [4,5]. When some reference frame (a physical object) is introduced and the system of coordinates (a mathematical object) is adopted in it, then F ab can be written as a coordinate-based-geometric quantity (CBGQ) containing components and a basis. The system of coordinates with the Einstein synchronization of clocks and Cartesian spatial coordinates (it will be called the Einstein system of coordinates (ESC)) is almost always chosen in the usual treatments. (In my approach to SR that uses 4D quantities defined without reference frames, [4,5] and [8] in the TF, and [6] in the CAF, any permissible system of coordinates, not necessary the ESC, can be used on an equal footing.) When F ab ab µν µν is written as a CBGQ (with the ESC) it becomes F = F γµ γν , where Greek indices µ, ν in F run from 0 to 3 and they denote the components of the geometric object⊗ F ab in some system of coordinates, here the ESC, γµ are the basis 4-vectors (not the components) forming the standard basis γµ and 0 0 denotes the tensor product of the basis 4-vectors. In the TF I shall often denote the unit 4-vector{ in} the time⊗ b direction γ0 as t as well. Then in some reference frame (with the ESC, i.e., with the standard basis γµ ) b b µ µ { } t canbealsowrittenasaCBGQ,t = t γµ,wheret is a set of components of the unit 4-vector in the time direction (tµ =(1, 0, 0, 0)). Almost always in the standard covariant approaches to SR one considers only the components of the geometric quantities taken in the ESC, and thus not the whole tensor. However

1 the components are coordinate quantities and they do not contain the whole information about the physical quantity. In the standard treatments one defines the sets of components of the electric and magnetic fields as µ µν µ µνλσ µν eqnarray E = F tν ,B=(1/2)ε Fλσ tν =(F ∗) tν , Let us now apply the LT to the components given in (ebmi). Under the passive LT the sets of com- µ µ µ µ ponents E and B from (ebmi) transform to E0 and B0 in the relatively moving IFR S0 eqnarray µ µν µ µνλσ µν E0 = F 0 vν0 ,B0 =(1/2)ε Fλσ0 vν0 =(F ∗)0 vν0 , In contrast to the above consideration in all usual treatments, e.g., [2,3] and [11] eqs. (3.5) and (3.24), in µν S0 one again simply makes the identification of six independent components of F 0 with three components i i i0 i i ikl E0 , E0 = F 0 , and three components B0 , B0 =(1/2)ε Flk0 . This means that standard treatments assume µ ν ν that under the passive LT the set of components t =(1, 0, 0, 0) from S transform to t0 =(1, 0, 0, 0) (t0 are the components of the unit 4-vector in the time direction in S0 and in the ESC), and consequently that µ µ µ µ µ µν µ µν µ E and B from (ebmi) transform to Est.0 and Bst.0 in S0, eqnarray Est.0 = F 0 tν0 ,Bst.0 =(F ∗)0 tν0 ; Est.0 = 0,E1,γE2 γβB3,γE3 + γβB2 , − From the relativistically incorrect transformations (kr) one simply derives the transformations of the i i i i spatial components Est.0 and Bst.0 . As can be seen from (kr) the transformations of Est.0 and Bst.0 are exactly the standard transformations of components of the 3-vectors E and B that are obtained by Einstein in [1] and subsequently quoted in almost every textbook and paper on relativistic electrodynamics. Then in the same way as in S the 3-vectors E0 and B0 as geometric quantities in the 3D space are constructed in S0 i i 10 20 30 from the spatial components Est.0 and Bst.0 and the unit 3-vectors i0,j0,k0, e.g., E0=F 0 i0 + F 0 j0 + F 0 k0. i i Both the transformations (kr) and the transformations for Est.0 and Bst.0 (i.e., for E0 and B0) are typical examples of the ”apparent” transformations (AT) that are first discussed in [9] and [10]. The AT of the spatial distances (the Lorentz contraction) and the temporal distances (the dilatation of time) are elaborated in detail in [4] and [8] (see also [12]), and in [4] I have discussed the AT of E and B. It is explicitly shown in [8] that the true agreement with experiments that test SR exists only when the theory deals with well-defined 4D quantities, i.e., the quantities that are invariant upon the passive LT. i i In all previous treatments of SR, e.g., [1-3] [11], the transformations for Est.0 and Bst.0 are considered to be the LT of the 3D electric and magnetic fields. However our analysis shows that the transformations for i i Est.0 and Bst.0 are derived from the relativistically incorrect transformations (kr) and that the 3-vectors E0 and B0 are formed by an incorrect procedure in 4D spacetime, i.e., by multiplying these relativistically incorrect components with the unit 3-vectors. All this together exactly proves that the standard transformations i i for E0 and B0 have absolutely nothing to do with the LT, and that the quantities Est.0 and Bst.0 , i.e., the 3-vectors E and B are not well-defined 4D quantities. Consequently the usual ME with 3D E and B are not in agreement with SR and they are not physically equivalent with relativistically correct FE with F ab (see also [4]). The relation (ebmi) reveals that we can always select a particular - but otherwise arbitrary - IFR S in which the temporal components of Eµ and Bµ are zero. Then in that frame the usual ME for the spatial components Ei and Bi (of Eµ and Bµ) will be fulfilled. As a consequence the usual ME can explain all experiments that are performed in one reference frame. However as shown above the temporal components µ µ of E0 and B0 are not zero; (ebcr) is relativistically correct, but it is not the case with (kr). This means that the usual ME cannot be used for the explanation of any experiment that test SR, i.e., in which relatively moving observers have to compare their data obtained by measurements on the same physical object. The relations (ebcr) imply that in the ESC the well-defined 4D electric and magnetic fields will be µ µν µ µν CBGQs, the 4-vectors, E γµ = F vν γµ and B γµ =(F ∗) vν γµ respectively. In an arbitrary chosen IFR Svν can be taken to be in the time direction, i.e., vν = tν , whence one finds (ebmi) in S and (ebcr) in any µ µν relatively moving IFR S0. (The components in the ESC, e.g., E = F vν , and the covariant formulation of electrodynamics with them, are considered in [13], [12] and [14]). In order to have the electric and magnetic 4-vectors defined without reference frames, i.e., independent of the chosen reference frame and of the chosen system of coordinates in it, we employ the abstract tensors and write Ea = F abv and Ba = (1/2)εabcdv F , b − b cd see [4,5] and [7]. The velocity vb and all other quantities entering into these relations are defined without a a reference frames. vb characterizes some general observer. Thus the relations for E and B hold for any observer. When some reference frame is chosen with the ESC in it and when vb is specified to be in the time direction in that frame, i.e., vb = tb, then all results of the classical electromagnetism are recovered

2 in that frame. However, in contrast to the description of the electromagnetism with the 3D E and B, the description with Ea and Ba is correct not only in that frame but in all other relatively moving frames and it holds for any permissible choice of coordinates. In [4] I have also presented the form of the LT that is independent of the chosen coordinates. Furthermore I have developed three equivalent but independent, consistent and complete formulations of electrodynamics with abstract tensors, with F ab, Ea and Ba, and with their complex combination. (Note that the above relations for Ea and Ba are not the physical definitions of Ea and Ba but they simply connect the independent and complete formulations with F ab and Ea,Ba. The physical definitions of Ea and Ba are given in terms of the Lorentz force expressed with Ea and Ba and Newton’s second law, as in [4] in the TF and in the first paper in [6] in the CAF.) The same situation as in TF happens in the standard CAF, e.g., [15,16]. The ESC, i.e., the standard basis γµ , is chosen from the outset and the relations for E and B are written explicitly using the basis 1- vector{ in} the time direction, γ .InsomeIFRS, E = F γ = F k0γ ,andB = γ (F γ )=(1/2)ε0iklF γ , 0 · 0 k − 5 ∧ 0 lk i where γ5 is the pseudoscalar for the frame γµ ,’’and’ ’ denote the inner and outer products of the basis 1-vectors. (This form for E and B is equivalent{ } to· the forms∧ given in the standard CAF, e.g., [15,16].) Then it is wrongly assumed that under the active LT (expressed in CAF by rotors; see [15,16], [6]) the new, i.e., the Lorentz transformed, E0 and B0 are E0 = F 0 γ and B0 = γ (F 0 γ ). However when E is transformed st. · 0 st. − 5 ∧ 0 by the active LT then R(F γ )R is not equal to (RFR) γ = E0 (for the explicit form of the rotor R · 0 e e · 0 st. see, e.g., [16] ch.6 and [6]). Really the components of Est.0 (Bst.0 ) are the same as in the AT (kr), while the components of RER (RBR) are the same as in the correct LT (ebcr). Finally the standard transformations e e for the 3D E and B are derived from the AT for Est.0 and Bst.0 , see [15] Sec. 18 and [16] Ch. 7. In [6] I have presented the relativistically correct form for E and B by replacing γ0 with v, the velocity of some general observer, which is also defined, as in TF, without reference frames. The CAF from [6] enabled me to develop four equivalent but independent, consistent and complete formulations of electrodynamics with field bivector F ,with1-vectorsE and B , with complex 1-vector Ψ and, what is specific for the CAF, with a real Clifford multivector Ψ. Furthermore all relevant quantities for the electromagnetism, the stress-energy 1-vector T (v), the energy density U (scalar), the Poynting 1-vector S, the angular momentum density M (bivector) and the Lorentz force K (1-vector) are directly derived from the FE and all of them are defined without reference frames. The whole consideration explicitly shows that the 3D quantities E and B, their transformations and the equations with them are ill-defined in the 4D spacetime. More generally, the 3D quantities do not have an independent physical reality in the 4D spacetime. Thence the relativistically correct physics must be formulated with 4D quantities that are defined without reference frames, or by the 4D CBGQs (e.g., as in [4,5], [8] in the TF and [6] in the CAF). The principle of relativity is automatically included in such theory with well-defined 4D quantities, while in the standard approach to SR [1] it is postulated outside the mathematical formulation of the theory. The comparison with experiments from [8] and [6] reveals that the true agreement with experiments that test SR can be achieved only when such well-defined 4D quantities are considered.

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